SOLUTION: Divide f (x) = 12x4 � 67x3 + 108x2 � 47x + 6 by p(x) = x � 2, to f ind the quotient. Use the remainder theorem to confirm you results

Question 113803: Divide f (x) = 12x4 � 67x3 + 108x2 � 47x + 6 by p(x) = x � 2, to f ind the quotient.
Use the remainder theorem to confirm you results

Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!
Let's simplify this expression using synthetic division

Start with the given expression

First lets find our test zero:

Set the denominator equal to zero

Solve for x.

so our test zero is 2

Now set up the synthetic division table by placing the test zero in the upper left corner and placing the coefficients of the numerator to the right of the test zero.

2	\|	12	-67	108	-47	6
	\|

Start by bringing down the leading coefficient (it is the coefficient with the highest exponent which is 12)

2	\|	12	-67	108	-47	6
	\|
		12

Multiply 2 by 12 and place the product (which is 24) right underneath the second coefficient (which is -67)

2	\|	12	-67	108	-47	6
	\|		24
		12

Add 24 and -67 to get -43. Place the sum right underneath 24.

2	\|	12	-67	108	-47	6
	\|		24
		12	-43

Multiply 2 by -43 and place the product (which is -86) right underneath the third coefficient (which is 108)

2	\|	12	-67	108	-47	6
	\|		24	-86
		12	-43

Add -86 and 108 to get 22. Place the sum right underneath -86.

2	\|	12	-67	108	-47	6
	\|		24	-86
		12	-43	22

Multiply 2 by 22 and place the product (which is 44) right underneath the fourth coefficient (which is -47)

2	\|	12	-67	108	-47	6
	\|		24	-86	44
		12	-43	22

Add 44 and -47 to get -3. Place the sum right underneath 44.

2	\|	12	-67	108	-47	6
	\|		24	-86	44
		12	-43	22	-3

Multiply 2 by -3 and place the product (which is -6) right underneath the fifth coefficient (which is 6)

2	\|	12	-67	108	-47	6
	\|		24	-86	44	-6
		12	-43	22	-3

Add -6 and 6 to get 0. Place the sum right underneath -6.

2	\|	12	-67	108	-47	6
	\|		24	-86	44	-6
		12	-43	22	-3	0

Since the last column adds to zero, we have a remainder of zero. This means is a factor of

Now lets look at the bottom row of coefficients:

The first 4 coefficients (12,-43,22,-3) form the quotient

So

You can use this online polynomial division calculator to check your work

Summary:

So our quotient is

Our remainder is zero

Since our remainder is zero,
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